Open subgroups and Pontryagin duality

Math. Z. 215, 195 204 (1994)

Mathematische Zeitschrift ~j Springer-Verlag 1994

Open subgroups and Pontryagin duality Wojciech Banaszczyk 1, Maria Jesfis Chasco 2, Elena Martin-Peinador 3,. Institute of Mathematics, L6d~ University, Banacha 22, PL-90-238 L6d~, Poland 2 Departamento de Matem'atica Aplicada, Universidad de Vigo, E.T.S. Ingenieros lndustriales, Apartado, 62, Vigo, Spain 3 Departamento de Geometrla y Topologia, Facultad de Matemfiticas, Universidad Complutense, E-28040 Madrid, Spain Received 21 February 1991; in final form 12 March 1992

0 Introduction By a character of a g r o u p G we mean a h o m o m o r p h i s m of G into the g r o u p R/Z. If G is an abelian topological group, the set of all its continuous characters, with addition defined pointwise and the c o m p a c t - o p e n topology, is a Hausdorff abelian g r o u p ; we call it the dual group or the character group of G and denote it by G . We say that G is reflexive if the evaluation m a p is a topological isomorphism of G onto G A^ Let A be an open s u b g r o u p of an abelian topological g r o u p G. V e n k a t a r a m a n [5] proved that if G is reflexive, then so is A (see, however, R e m a r k 2.4 below). Under certain additional assumptions, this result had been obtained earlier by Noble [4, Corollary 3.4]. In Sect. 2 of the present paper we show that the reflexivity of G is, in fact, equivalent to the reflexivity of A. We also deal with the relationship between the reflexivity of the groups G and G/K where K is a c o m p a c t s u b g r o u p of G. An abelian topological g r o u p G is called strongly reflexive if all closed subgroups and Hausdorff quotient groups of G and G are reflexive. This notion was introduced in [2] where countable products of lines and circles were investigated (cf. R e m a r k 3.2 below). The class of strongly reflexive groups comprises, among other things, nuclear Fr6chet spaces and countable products of locally compact abelian groups [-1, (l 7.3)]. M o r e information on strong reflexivity can be found in [1, Sect. 17]. Let G be an abelian topological group. Let A be an open and K a c o m p a c t subgroup of A. In Sect. 3 we prove that if A is strongly reflexive, then so is G. Furthermore, if G/K is strongly reflexive and G admits sufficiently m a n y continuous characters (i.e. if continuous characters separate points of G), then G is strongly reflexive, too. The converse statements are also true" see (3.1.d). * Partially supported by D.G.I.C.Y.T. grant BE91-031

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1 Preliminaries

The group R / Z will be denoted by T. It is convenient to identify T with the interval ( - 89 89 and, consequently, to treat characters as real-valued functions. Let S be a subset of an abelian topological group G. By gp S we denote the subgroup of G generated by S. Given a character X of G, we shall write [z(S)I = sup {[z(g)l" g~ S}. The set

{zeG ^ Iz(S)I=1} is called the polar of S; we shall denote it by S o. If S is a subgroup of G, then S o is a closed subgroup of G; it consists of all characters of G which vanish on S. (1.1) Lemma. The polars of compact subsets of an abelian topological group G form a basis of neighbourhoods of zero in G A. The easy proof is left for the reader. (1.2) Lemma. I f U is a neighborhood of zero in an abelian topological group G, then U ~ is a compact subset of G ^. This is a standard fact; see e.g. [4, L e m m a 2.2]. Let H be a closed subgroup of an abelian topological group G. We say that H is dually closed in G if to each g ~ G \ H there corresponds a character z ~ G ~ with ZIn---0 and z(g)+0. Next, H is said to be dually embedded in G if each continuous character of H can be extended to a continuous character of G. The canonical h o m o m o r p h i s m s G ~ / H ~ A and (G/H)A~ H ~ defined in the obvious way, are denoted by q~/ and q~u, respectively. Notice that q~n is a continuous injection; it is surjective if and only if H is dually embedded in G. The mapping q~u is a continuous isomorphism. The evaluation m a p of G into G ^^ is denoted by ~a. The verification of the following simple fact is left to the reader: (1.3) Lemma. I f H is a dually closed subgroup of G, then ~G(H)= H~176 ~ c~a(G). Let G, H be abelian topological groups and ~, : G ~ H a continuous h o m o m o r p h ism. The dual h o m o m o r p h i s m ~9A: HA--.G ^ is defined by ~ ^ ( Z ) = X ~ , xeHA; it is clear that t)A is continuous. A direct verification shows that the diagram

G

~^ is commutative.

~

r

,

H

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(1.4) L e m m a . Let F, G, H be abelian topological groups. Let 49: F ~ G and t~: G ~ H be continuous homomorphisms such that the sequence 0

)F

4, , G

q' ) H

)0

is exact. I f t~ is open, the sequence FA* 4,-

G^,

~

H~

0

is exact. If, in addition, 49 is open, then 0 ^ maps G ^ onto F ^. Proof We have k e r O ^ = { 0 } because O ( G ) = H . We shall prove that ker49 A = i m O A. Each character Z belonging to ker49 A vanishes on 49(F), hence on ker ~. Since t9 (G) = H, it follows that there is a unique h o m o m o r p h i s m ~c: H --* T such that Z = X O . As Z is continuous and ~ open, x is continuous. We have O ^ ( x ) = ~ c O = Z. This proves that ker 4 9 - ~ i m A . The opposite inclusion follows from the equalities 49^ ~ ^ = (~9 49) ^= 0 A= 0. The last assertion of the l e m m a follows, for instance, from the fact that open subgroups are dually e m b e d d e d (cf. (2.2.b)). []

(1.5) L e m m a . Suppose we are given a commutative diagram F

4, , G

q' ~ H

F'

4,' , G'

0'

~H'

of abelian groups and their homomorphisms. Suppose further that c~ and 7 are isomorphisms. I f im ~ = H and ker ~' = i m 49', then im fl = G'. / f ker 49' = {0} and ker ~b= im 49, then ker fl = {0}.

The p r o o f consists in a direct verification. (1.6) L e m m a . Let G, H be Hausdorff groups (abelian or not) and let ~: G--* H he a continuous homomorphism. Suppose that ~9 is open and its kernel is compact. Then the Jbllowing statements are true: (a) Let (gi) be a net in G; if the net ((J(gi)) has a cluster point in H, then (gi) has a cluster point in G. (b) ~ is a closed mapping. (c) The inverse images of compact subsets of H are compact subsets of G. Proof Statements (b) and (c) are standard. Besides, they follow easily from (a). We shall prove (a). Let % and en denote the neutral elements of G and H, respectively. We may assume that en is a cluster point of (O(g/)). We shall prove that (gl) has a cluster point in K , = k e r ~. Suppose the contrary. Then to each p e K there correspond an index ip and an open n e i g h b o u r h o o d Up of p in G, such that gi(~Up for i>=ip. As K is compact, the open covering {Up}pc K of K has a finite

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subcovering {Up}p~s. Then U = ~

Up is an open subset of G containing K,

peS

and giCU for all sufficiently large i, say, for i>io. As K is compact, there is a neighbourhood V of ea with K V c U . Then 0(V) is a neighbourhood of en, and 0(gi)r for i>io, contrary to our assumption that et~ is a cluster point of (0(gi)). []

2 Open subgroups and duality (2.1) Lemma. Let G, H be abelian topological groups and 0 : G --* H a continuous homomorphism. Suppose that 0 is open, maps G onto H, and that ker 0 is compact. Then the dual homomorphism 0 ^ : H^--* G ^ is open.

The assumption that 0 (G)= H may be dropped; cf. Lemma 2.5 below. Proof. Take an arbitrary compact subset X of H. In view of (1.1), it is enough

to show that 0 ^ ( X ~ is a neighbourhood of zero in G ^. Denote K = k e r 0. It follows from (1.6.c) that K u 0 - 1 ( X ) is compact. Now, it is not hard to see that ( K w O - ~ ( X ) ) ~ c O ^ ( X ~ [] (2.2) Lemma. Let A be an open subgroup of an abelian topological group G. Consider the canonical commutative diagram 0

,

A

0

, A ^^

u

,

G

~ ,

G/A

,0

, (G/A)

,0.

(*) u

, G

~

Then the following assertions are true: A is dually closed in G; A is dually embedded in G; A ~ is a compact subgroup of G^; p ^ ' , G ^ ~ A iLs open and surjective; (e) # 9 A -~ G is open and injective; (f) (aA" G ^ / A ~ --* A A is a topological isomorphism; (g) ~bA:(G/A) ^-~ A ~ is a topological isomorphism; (h) both rows in diagram (*) are exact; (i) A ~ is dually embeded in G~; (j) A ~ is dually closed in G ^.

(a) (b) (c) (d)

P r o o f The discrete group G/A admits sufficiently many continuous'characters, which proves (a). For (b), see [4, Lemma 3.3]. Statement (c) follows from (1.2). Now, (b) says that # " G ---,A^ is surjective. To prove that #~ is open, take an arbitrary compact subset X of G. In view of (1.1), we only have to show that ~L"(X~ is a neighbourhood of zero in A ."As v(X) is finite, the group C = g p v ( X ) is a direct sum of some cyclic subgroups C1 ..... C,. For each k = l , . . . , n , choose a generator c k of Ck and then some gkev-l(ck); let Sk bc the order of Ck.

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Let us denote D = gp {gk}~,=~, I = {k: Sk < O0} and Q = {pl gl + . . . + p . gn: Pl . . . . . pnffZ a n d O~=Pk